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F-tests continued  Two kinds of F-tests  Overall F-test (or Global F-test) tests whether or not there is a regression relation between Y and the set of covariates For a regression with p covariates, the overall F-test compares F* = MSR/MSE ~ F(p, n-p-1)

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Partial F test  partial because it tests “part” of the model.  tests one or more covariates simultaneously  Can be done using the ANOVA table, if covariates are entered in the ‘correct’ order  Or, by comparing results from regression tables  Examples:

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F-tests and p-values in ANOVA table  They are tests for a covariate, conditional on what is above it in the table.  Example: F statistic for INFRISK tests is it adjusted for other covariates?  no  it tests INFRISK in the presence of no other covariates  p < 0.0001

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Implications  ANOVA table results depends on the order in which the covariates appear  If you want to use ANOVA table to test one or more covariates, they should come at the end  reg1: we can see if INFRISK is significant without any adjustments we can see if nurse2 is significant adjusting for everything else  reg1a: we can see if INFRISK is significant adjusting for everything else we can see if nurse2 is significant, adjusting for NURSE and ms, but not adjusting for INFRISK

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F-tests (continued)  Partial F-test for >1 covariate  Implications: The denominator is always the MSE from the full model The numerator can always be determined by entering the covariates in the order in which you want to test them Recall: additivity of sums of squares

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Concluding remarks r.e. F-test  Global F-test: not very common, except for very small models  Partial F-test for individual covariate: not very common because it is the same as the t-test  Partial F-test for set of covariates: quite common easiest to find ANOVA table for nested models can use ANOVA table from full model to determine F- statistic

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Coefficient of Determination  Also called R 2  Measures the variability in Y explained by the covariates.  Two questions (and think ‘sums of squares’ in ANOVA): How do we measure the variance in Y? How do we measure the variance explained by the X’s?

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R2R2  The coefficient of determination is defined as SST: Variance in Y SSR: Variance explained by X’s SSE:Variance left over, not explained by regression

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Use of R 2  Similar to correlation  But, not specific to just one X and Y  Partitioning of explained versus unexplained  For certain models, it can be used to determine if addition of a covariate helps ‘predict’

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Misunderstandings r.e. R 2  A high R 2 indicates that a useful prediction can be made there still may be considerable uncertainty, due to small N. recall that predictions depend on how close “X” is to the mean  A high R 2 indicates that the regression model is a ‘good fit’ high R 2 says nothing about adhering to model assumptions standard diagnostics should still be used, even if R 2 is high  R 2 near 0 indicates X and Y are not related. you can still have strong association with a lot of unexplained variance (e.g., age and cancer) for similar reasons as above, need to look at modeling X and Y may be related, but not linearly

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R 2 decreased?  The addition of a covariate will ALWAYS increase the R 2 value.  Why? there is always at least a little bit explained by the new X the only possible way to have no increase in R2 would be if the addition of the new covariate had estimated β = 0 It is ‘almost never’ true that the slope estimate is exactly.  Extreme case: perfect linear association between two covariates (e.g., age in years and age in months)

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“Solution”  Adjusted R  Accounts for the number of covariates in the model  “Purists” do not like the adjusted R 2  The adjusted only increases with a new covariate if the new term “improves” the model more than expected by chance alone.

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Coefficients of Partial Determination  Measures the marginal contribution of one X variable when all others are already in the model  Intuitively, how much variation in Y are we explaining, after accounting for what is already in the model?  Construction in Two Covariate case: